Finite Group Databasebeta

← #27302750

Subgroups — #27302750

Order 13568.

488 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274707C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20272833C214yesmaximal
4C53×#2487C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133134C2212yes—
7C2×#13133132C2212yes—
8C4×C22⋊Dic53—21no—
9#16810392—21no—
10#16810390—21no—
11C2×C4×D4×C53C2211yescharacteristic
12C22×C4⋊C4×C53C411yescharacteristic, complemented
13C53×#400—21no—
14C53×#399—21no—
15C53×#389—21no—
16C53×#386C2212yes—
17C53×#385C412yescomplemented
18C53×#380—21no—
19C53×#380C411yescharacteristic, complemented
20C2×#11719917—22no—
21C2×C22⋊Dic53—22no—
22C2×C4×Dic53—41no—
23C4⋊#88322—41no—
24C4⋊#88322—41no—
25#13133134—24no—
26#13133132—22no—
27#13133132Q814yes—
28#13133124—41no—
29C22×D4×C53C2311yescharacteristic
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2×C412yes—
32C23×C4×C53D411yescharacteristic
33C4×D4×C53—41no—
34C4×D4×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—41no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53C2×C412yes—
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—22no—
48#95×C53—21no—
49#95×C53—22no—
50C24×C53C22×C411yescharacteristic
51C24×C53C2×D411yescharacteristic
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C2×Q812yes—
55C2×D4×C53#4112yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—82no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—82no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—82no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—24no—
85C22⋊Dic53—42no—
86C4×Dic53—82no—
87#11719917—42no—
88#11719917—42no—
89C2×#88322—42no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53C4×Q812yes—
100C23×C53C22⋊Q812yes—
101D4×C53—42no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53—84no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#12411yescharacteristic
126C2×C4×C53C4⋊Q811yescharacteristic
127C2×Dic53—44no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130#88322—82no—
131#13295—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53#39111yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53#41011yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—82no—
154C4×C53—82no—
155C4×C53#35212yes—
156Dic53—84no—
157#2487D5311yescharacteristic
158#1800—531no—
159C4×#99—531no—
160#1390—534no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—42no—
164C2×C53—42no—
165C2×C53#199611yescharacteristic
166C2×C53C2×#35211yescharacteristic
167C2×C53#212211yescharacteristic
168C2×C4×D4C2×D5311yescharacteristic
169C22×C4⋊C4Dic5311yescharacteristic, complemented
170#423—1061no—
171C2×#100—532no—
172C2×#99—532no—
173#400—21no—
174#399—21no—
175#389—21no—
176#386C2×D5312yes—
177#385Dic5312yescomplemented
178#380—21no—
179#380Dic5311yescharacteristic, complemented
180C4×C22⋊C4—1061no—
181#343—1061no—
182#342—1061no—
183C53#1329511yescharacteristic, complemented, Sylow
184C22×D4C22×D5311yescharacteristic
185C23×C4C2×Dic5311yescharacteristic
186C23×C4C2×Dic5312yes—
187C23×C4C22⋊D5311yescharacteristic
188C2×M16—1062no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4C2×Dic5312yes—
201C2×C22⋊C4C2×Dic5311yescharacteristic
202C2×C22⋊C4C22⋊D5311yescharacteristic
203C2×C42—2121no—
204C2×C42C22⋊D5311yescharacteristic
205C4⋊C8—2121no—
206C4⋊C8—2121no—
207#100—1064no—
208#99—534no—
209#99—1062no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—2121no—
214C24C22×Dic5311yescharacteristic
215C24C2×C22⋊D5311yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4Q8×D5312yes—
219C2×D4#1171994912yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—2122no—
231C22×C4C22⋊Dic5311yescharacteristic
232C22×C4C22⋊Dic5311yescharacteristic
233C22×C4C22⋊Dic5311yescharacteristic
234C22×C4C22⋊Dic5311yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236M16—2122no—
237M16—2122no—
238C2×C8—2122no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—82no—
249C22⋊C4—1064no—
250C22⋊C4—2122no—
251C42—21no—
252C42—21no—
253C42—4242no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊Dic5311yescharacteristic
263C23Q8×Dic5312yes—
264C23#1313319312yes—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—84no—
287C2×C4—4242no—
288C2×C4—4242no—
289C2×C4—2124no—
290C2×C4#1313313211yescharacteristic
291C2×C4#1313313211yescharacteristic
292C2×C4#1313320611yescharacteristic
293C2×C4#1313321511yescharacteristic
294C8—4242no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1681053011yescharacteristic
308C22C2×#1313313211yescharacteristic
309C22#1681059711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—82no—
318C4—4244no—
319C4#1681041312yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2027570111yescharacteristic
325C2C2×#1681041311yescharacteristic
326C2#2027636511yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2487C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2487D53
hypocenterC53#13295
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. C22×C4⋊C4×C53C4semidirect
17. C53×#385C4semidirect
19. C53×#380C4semidirect
169. C22×C4⋊C4Dic53semidirect
177. #385Dic53semidirect
179. #380Dic53semidirect
183. C53#13295semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202747072
C4×#131331322
#202728332
C53×#24872
#1329553